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This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator). However, the following simulation results show that the optimal variable parameters are different for different fractional orders. Then the weighted square integral index about the magtitude and phase is defined as the objective functions to achieve the optimal variable parameter for different fractional orders. Finally, the simulation results demonstrate that there are great differences on the optimal variable parameter for differential and integral operators with different fractional orders, which should be attracting more attentions in the design of digital fractional order controller.

Fractional order calculus has a history of more than 300 years, which extends the order of the classical calculus from integer number to arbitrary real number and even complex number. Compared with integral order calculus, the fractional order calculus could describe the dynamic characteristics of the actual system more accurately. Therefore, fractional order control is increasingly becoming one of the most important topics in control theory in recent years [

The discretization of the fractional-order differentiator and integrator is the foundation of the digital realization of fractional order controller. Generally, there are two methods for the discretization of the fractional-order differentiator and integrator [

The discretization of the fractional-order differentiator is taken for example. The direct discretization method could be summarized as the following two steps. Firstly, some kind of generating function ω ( z − 1 ) is used to discretize the differentiator s , i.e., s r = ω r ( z − 1 ) , where r is the order of the fractional- order differentiator and ω ( z − 1 ) is usually expressed as a function of the com- plex variable z or the shift operator z − 1 , and then some kind of expansion method is applied to generate the approximate digital filter of the differentiator. For example, Chen and Vinagre propose an IIR (infinite impulse response)-type digital fractional-order differentiator with weighted sum of Simpson integration rule and the trapezoidal integration rule [

The remainder of this paper is organized as follows. The preliminary of frac- tional calculus is briefly introduced in Section II. The discretization of fractional- order operator with different orders is discussed in Section III. Finally, the con- clusions are given in the last section.

A. Definition of Fractional Calculus

Fractional order calculus is a natural generalization of the classical integral order calculus, which extends the order of the integration and differentiation to the non-integer or fractional order [

a D t r f ( t ) = lim h → 0 h − r ∑ j = 0 [ t − a h ] ( − 1 ) j ( r j ) f ( t − j h ) , (1)

where a D t r is the fractional-order calculus operator, a and t are the limits of the operator respectively, r is the order of the operator and [ ⋅ ] means the integer part. The RL definition is defined as

a D t r f ( t ) = 1 Γ ( n − r ) d n d t n ∫ a t f ( τ ) ( t − τ ) r − n + 1 d τ , (2)

where, n − 1 < r < n and Γ ( ⋅ ) is the Gamma function. Actually, the afore- mentioned definitions are equivalent to each other in the real physical systems and engineering applications.

The Laplace transform of fractional order calculus with zero initial conditions for order r is

L { a D t r f ( t ) } = s r F ( s ) , (3)

where L { ⋅ } denotes the operation of Laplace transform, F ( s ) is the Laplace transform of function f ( t ) and s r denotes the fractional-order operator [

B. Parameterized Al-Alaoui Transform

The discretization of the fractional-order differentiator and integrator is to design a digital filter for the fractional-order operator s r . Firstly, a generating function is used to realize the transform of the Laplace operator from the continuous complex frequency domain to the discrete complex frequency domain. According to the definition of Z transform, the mapping relationship between the two domains is

z = e s T , (4)

where T is the sampling period. Furthermore, the equivalent relation can be formulated

z = e s T = e s ( ( 1 − α ) T + α T ) = e ( 1 − α ) T s e − α T s , (5)

where α ∈ [ 0 , 1 ] . Take power series expansion to the numerator and denomi- nator of Equation (5) and neglect the high-order terms, Equation (5) can be approximated as

z = ∑ n = 0 ∞ [ ( 1 − α ) T s ] n / n ! ∑ k = 0 ∞ ( − α T s ) k / k ! ≈ 1 + ( 1 − α ) T s 1 − α T s , (6)

and then the complex variable s can be solved to yield

s = f ( z , α ) = 1 T z − 1 1 + α ( z − 1 ) . (7)

Equation (7) is defined as the α transform from continuous complex frequency domain to the discrete complex frequency domain in [

s = 1 T z − 1 1 + α ( z − 1 ) = 2 ( z − 1 ) T [ ( 1 − a + ( 1 + a ) ) z ] . (8)

Specially, three commonly used generating functions, i.e., Tustin transform, Al-Alaoui transform and Euler transform can be formulated when a is set to 0, 3/4 and 1, respectively.

A digital integrator with adjustable parameters, which is used in the discreti- zation of fractional-order operator [

s = 1 β T 1 − z − 1 γ + ( 1 − γ ) z − 1 , (9)

where β and γ are gain adjusting parameter and phase adjusting parameter, respectively. The fractional-order operator can be adjusted more accurately with the adjusting parameter according to different real applications. Specially, Equation (9) is equivalent to Equation (7) when parameter β is set to 1. Additionally, if the parameter α in Equation (7) is set to 1 / ( 1 + δ ) , the generating function is formulated

s = 1 + δ T 1 − z − 1 1 + δ z − 1 , (10)

which is used for the discretization of fractional-order operator in [

From above analyses, the generating functions in Equations (7), (8), (9) and (10) are actually equivalent to each other when certain relations of the variable parameters are satisfied as shown in

C. Power Series Expansion of Generating Function

Two main methods for the expansion of generating function are continued fractional expansion and power series expansion. Generally, CFE method could

Transform | |||||
---|---|---|---|---|---|

1 | 1 | 1 | 0 | Euler | |

7/8 | 3/4 | 7/8 | 1/7 | Al-Alaoui | |

1/2 | 0 | 1/2 | 1 | Tustin |

obtain IIR-type digital filter which is easy for the digital filter design, while PSE method could obtain FIR (finite impulse response)-type digital filter and re- quires less computation cost in comparison with CFE under similar accuracy criterion [

In this paper, Equation (10) is used as the generating function, and the power series expansion algorithm presented in [

D r ( z − 1 ) ≈ PSE { ω r ( z − 1 ) } p , q ≈ P p ( z − 1 ) Q q ( z − 1 ) ≈ PSE { ( 2 T 1 − z − 1 1 + z − 1 ) r } p , q = ( 2 T ) r PSE { ( 1 − z − 1 ) r } p PSE { ( 1 + z − 1 ) r } q = ( 2 T ) r p 0 + p 1 z − 1 + ⋯ + p n z − n q 0 + q 1 z − 1 + ⋯ + q n z − n (11)

where, D r ( z − 1 ) is the digital filter of the discrete fractional-order operator, PSE { ⋅ } indicates the power series expansion, P p ( z − 1 ) and Q q ( z − 1 ) are the numerator and denominator polynomials of the digital filter, and p and q are their orders respectively. Without loss of generality, the approximate orders of the numerator and denominator polynomials are set to n (see [

A. Objective Function

Al-Alaoui transform has better properties in the discretization of fractional- order operator in comparison with other common generating functions, which have been reported in several papers (see e.g. [

However, the papers mentioned above usually consider certain specific fractional order (e.g. fractional order 0.5) and are not concerned with the discretization of fractional-order operator with different orders, which will influence the design of the digital filter. In this paper, we define the following objective functions

J mag = 1 ω b ∫ ω l ω u ‖ M c ( j ω ) − M d ( j ω ) ‖ 2 d ω J arg = 1 ω b ∫ ω l ω u ‖ A c ( j ω ) − A d ( j ω ) ‖ 2 d ω min J = w t ⋅ J mag + ( 1 − w t ) ⋅ J arg (12)

where M c ( j ω ) and M d ( j ω ) are the magtitude responses of the fractional- order differ-integrator and its discrete counterpart; A c ( j ω ) and A d ( j ω ) are the corresponding phase responses; ω b is the bandwidth betweeen the lower and upper limits ω l and ω u (i.e. ( ω b = ω u − ω l ) ) within a chosen frequency band, e.g. ω ∈ [ 10 − 1 , ω N ] with Nyquist frequency ω N . The normalized J mag and J arg depend on not only the parameter δ , but also the different orders r . The following simulation analyses are to find the optimal parameter δ for different fractional orders r , which could achieve the minimal objective function J with specific weight w t .

B. Simulation Results

In the simulation, the weight w t is taken as 0.75 without loss of generality. The orders of the power series expansion are taken as 5 for simplicity, the sampling period is taken as 0.001, and the orders of the fractional-order operator are typically taken as 0.1, 0.5, 0.9 and −0.1, −0.5, −0.9 for differential and integral operators respectively.

Figures 2-4 are the objective functions for differential operator with fractional order 0.1, 0.5 and 0.9, where the horizontal coordinate indicates the different

variable parameter δ and the longitudinal coordinates indicate the variation of J mag , J arg and J respectively. Figures 5-7 are the counterparts of the integral operator. All the figures demonstrate that the variation trends of the objective functions with variable parameter δ seem consistent with each other, while the optimal variable parameters are totally different for fractional differential and integral operator with different fractional orders under the selected objective functions.

Fractional Order | Optimal | Optimal |
---|---|---|

Differential Operator | Integral Operator | |

0.1 | 0.40 | 0.33 |

0.5 | 0.52 | 0.28 |

0.9 | 0.44 | 0.17 |

This paper is concerned with the discretization of the fractional-order differen- tiator and integrator with different fractional orders, which is seldom considered in previous literatures. The parameterized Al-Alaoui transform with one variable parameter is presented as a general generating function, and the objective functions are defined to achieve the optimal variable parameter for the discreti- zation. The simulation results demonstrate that there are great differences on the optimal variable parameters for the discretization of differentiator and integrator with different fractional orders.

However, the weight in the simulation is set as 0.75 without loss of generality, and it is undoubtedly arbitrary to select the proper weight for specific discreti- zation purpose, e.g. select smaller weight for more accurate phase approximation. In the future, we will consider the optimal variable parameter into the practical digital fractional order controller design to acquire the optimal control perfor- mances.

This work is supported by a Project of Shandong Province Higher Educational Science and Technology Program under Grant J14LN34.

Zhang, Q., Song, B.Y., Zhao, H.D. and Zhang, J.S. (2017) Dis- cretization of Fractional Order Differentiator and Integrator with Different Fractional Orders. Intelligent Control and Automation, 8, 75-85. https://doi.org/10.4236/ica.2017.82006